LIBRARY OF CONGRESS. 



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UNITED STATES OF AMERICA. 



A COMPLETED NOMENCLATURE FOR THE PRINCIPAL ROULETTES. 



F. N. WILLSON. 



A COMPLETED NOMENCLATURE 



FOR THE 



PRINCIPAL ROULETTES 



€ 




BY y^ 

FREDERICK NEWTON WILLSON. C.E. 

Prufemor of Descriptive Oeoniet/y, dterentoviii and Technical Drawing, 
Princeton College. New Jersey. 




Read before the 

American Association for the Advancement op Science, 

New York, August, 1887. 



!::i^ 



-\ 



JCs- 



/;■ / 



Copyright, 1887, 

by F. N. WILLSON. 






A COMPLETED NOMENCLATURE FOR THE PRINCIPAL ROULETTES. 



That would be an ideal nomenclature in which, from the etymology of 
the terms chosen, so clear an idea could be obtained of that which is 
named as to largely anticipate definition, if not, indeed, actually to render 
it superfluous. This ideal, it need hardly hardly be said, is seldom realized. 
As a rule we meet with but few self-explanatory terms, and the greater their 
lack of suggestiveness the greater the need of clear definition. Instances 
are not wanting of ill-chosen terms and even actual misnomers having 
become so generally adopted, in spite of an occasional protest, that we can 
scarcely expect to see them replaced by others more approj^riate. Whether 
this be the case or not, we have a right to expect, especially in the exact 
sciences, and jsreeminently in Mathematics, such clearness and comprehen- 
siveness of definition as to make ambiguity impossible. But in this we are 
frequently disappointed, and notably so in the class of curves we are to 
consider. 

Toward the close of the seventeenth century, the mechanician De la 
Hire gave the name of Roulette — or roll-traced cui-ve — to the path of a 
point in the j^lane of a curve rolling upon any other curve as a base. This 
suggestive term has been generally adopted, and we may expect its comple- 
mentary, and equally self-interpreting term, Grlissette, to keep it company 
for all time. 

By far the most interesting and important roulettes are those traced by 
points in the plane of a circle rolling uj^on another circle in the same plane, 
such curves having valuable practical applications in mechanism, while 
their geometrical properties have for centuries furnished an attractive field 
for investigation to mathematicians. 

The Terms Cycloids and Trochoids are somewhat indiscriminately used 
as general names for this class of curves. As far as derivation is concerned 
they are equally appropriate, the former being from kv/cXos, circle, and ei8os. 



form ; and the latter from rpo^os, wheel, and elSos. Preference has, however, 
been given to the terra Trochoids by several recent writers on mathematics 
or mechanism, among them Prof. E. H. Thurston and Prof. De Volson 
"Wood ; also Prof. A. B. W. Kennedy of England, perhaps best known in 
this country as the translator of Reuleaux' Theoretische Kinematik. Adopt- 
ing it for the sake of aiding in establishing uniformity in nomenclature I 
give the following definition : — 

If two circles are tangent, either externally or internally, and while one 
of them remains fixed the other rolls upon it without sliding, the curve described 
by any point on a radius of the rolling circle, or on a radius produced, will be 
a Trochoid. 

Of these curves the most interesting, both historically and for its math- 
ematical properties, is the cycloid, with which all are familiar as the i>ath of 
a point on the circumference of a circle which rolls upon a straight line, (i.e., 
the circle of infinite radius). 

The term "cycloid," alone, for the locus described, is almost uni- 
versally emjjloyed, although it is occasionally qualified by the adjectives 
right or common. 

Of almost equally general acceptation, also, although frequently inap- 
propriate,* are the adjectives curtate and prolate, to indicate trochoidal 
curves traced by points respectively without and vnthin the circumference of 
the rolliug circle (or generator as it will hereafter be termed) whether it roll 
upon a circle of finite or infinite radius. 

As distinguished from curtate and prolate forms all the other trochoids 
are frequently called common. 

Should the fixed circle (called either the base or director) have an infi- 
nite radius, or, in other words, be a straight line, the curtate curve is called 
by some the curtate cycloid; by others, the curtate trochoid, and similarly 
for the i^rolate forms. Since uniformity is desirable I have adopted the 
terms which seem to have in their favor the greater number of the authori- 
ties consulted, viz., curtate and prolate trochoid. It should also be further 
stated here, with reference to this word " trochoid," that it is usually the 
termination of the name of every curtate and j^rolate form of trochoidal 
curve, the termination cycloid indicating that the tracing point is on the cir- 
cumference of the generator. 

The general name Linear Trochoids, adopted by Prof. Kennedy for the 
curves having a straight line director, is suggestive and appropriate. 

With the base a straight line, the curtate curves consist of a series of 
loops, while the prolate forms are sinuous, like a wave line ; and the same 



* See page 14, first paragraph. 



is frequently true when the base is a circle of finite radius. Hence the sug- 
gestion of Prof. Clifford that the terras looped and luavij be employed instead 
of curtate and prolate. But we shall see, as we proceed, that they would not 
be of universal applicability, and that,excei>t with a straight line dii-ector, both 
curtate and prolate curves may be, in form, looped, wavy, or neither.* And 
we would all agree with Prof. Kennedy that as substitutes for these terms 
"Prof. Cayley's kru-noclal and ac-nodal hardly seem adapted for popular 
use." It is therefore futile to attempt to secvire a nomenclature which 
shall, throughout, suggest both the form of the locus and the mode of its 
construction, and we must rest content if we completely attain the latter 
desideratum. 

We have next to consider the trochoids traced during the rolling of a 
circle upon another circle of finite radius. At this point we find inadequacy 
in nomenclatui-e, and definitions involving singular auomalies. The earlier 
definitions have beeen summarized as follows by Prof. R. A. Proctor, in his 
valuable Geometry of Cycloids : — 

"The K ,-P ^ , ■-,{ is the curve traced out bv a point in the cir- 
( hypocycloid ) • '■ 

cumference of a circle which rolls without sliding on a fixed circle in the 

same plane, the two circles being in ^ . , i ^ contact." 
^ I internal ) 

As a specific example of this class of definition I quote the following 
from a more recent writer: — " If the generating circle rolls on the circum- 
ference of a fixed circle, instead of on a fixed line, the curve generated is 
called an epicycloid if the rolling circle and the fixed circle are tangent 
externally, a hypocycloid if they are tangent internally." (Byerly, Differen- 
tial Calculus, 1880.) 

In accordance with the foregoing definitions every epicycloid is also a 
hypocycloid, while only some hypocycloids are epicycloids. Salmon, (Higher 
Plane Curves, 1879) makes the followingexplicit statement on this point : — 
" Tbe hypocycloid, when the radius of the moving circle is greater than that 
of the fixed circle, may also be generated as an epicycloid." 

To avoid any anomaly, Prof. Proctor has presented the following unam- 
biguous definition: — 

" An ] ij .1 -^ .' is the curve traced out by a point on the cir- 

cumference of a circle which rolls without sliding on a fixed circle in the 
same plane, the rolling circle touching the ] . _. ■, - of the fixed circle." 



* See page 14, first paragraph. 



6 

This certainly does away with all confusion between the ejn- and hypo- 
curves, but we shall find it inadequate to enable us, clearly, to mate certain 
desirable distinctions. 

By some writers the term external epicycloid is used when the generator 
and director are tangent externally, and, similarly, internal epicycloid when 
the contact is internal and the larger circle is rolling. Instead of internal 
epicycloid we often find external hypocycloid used. It will be sufficient, 
with regard to it, to quote the following from Prof. Proctor: — " It has 
hitherto been usual to define it (the hypocycloid) as the curve obtained 
when either the convexity of the rolling circle touches the concavity of the 
fixed circle, or the concavity of the rolling circle touches the convexity of the 
fixed circle. There is a manifest want of symmetry in the resulting classifi- 
cation, seeing that while every epicycloid is thus regarded as an external 
hypocycloid, no hypocycloid can be regarded as an internal epicycloid. 
Moreover, an external hypocycloid is in leality an anomaly, for the prefix 
' hypo ' used in relation to a closed figure like the fixed circle, implies interi- 
orness." To avoid the confusion which it is evident, from the foregoing, has 
existed, and at the same time to conform to that principle which is always 
a safe one and never more important than in nomenclature, viz., not to use 
two words where one will suffice, I prefer reserving the term " epicycloid " for 
the case of external tangency, and substituting the more rec-ently suggested 
nsLxne pericycloid for both " internal epicycloid " and " external hypocycloid." 
The curtate and prolate forms would then be called peritrochoids. By the 
use of these names and those to be later presented we can easily make dis- 
tinctions which, without them, would involve undue verbiage in some cases, 
and, in others, the use of the ambiguous or inappropriate terms to which 
exception is taken. And the necessity for such distinctions frequently 
arises, especially in the study of kinematics and machine design. Take, for 
example, problems like many in the work of Keuleaux already mentioned, 
relating to the relative motion of higher kinematic pairs of elements, the 
centroids being circular arcs and the point-jsaths trochoids. In such cases 
we are quite as much concerned with the relative position of the rolling 
and fixed circles as with the form of a point-path. In solving problems in 
gearing the same need has been felt of simple terms for the trochoidal profiles 
of the teeth, which should imply the method of their generation. 

Although they have not, as yet, come into general use, the names peri- 
cycloid,and peritrochoid appear in the more recent editions of Weisbach and 
Reuleaux and will undoubtedly eventually meet with universal acceptance. 

Yet strong objection has been made to the term " pericycloid " by no 
less an authority than the late eminent mathematician. Prof. W. K. Clif- 
ford, who nevertheless adopts the " peritrochoid." I quote the following 
from his Elements of Dynamic : — " Two circles may touch each other so 



that each is outside the other, or so that one iacludes the other. In the 
former case, if one circle rolls upon the other, the curves traced are called 
epicycloids and epitrochoids. In the latter case if the inner circle roll on the 
outer the curves are hvpocycloids and hypotrochoids, but if the outer circle 
roll on the inner, the curves are ejjicycloids and peritrochoids. We do not 
want the name pericycloids, because, as will be seen, every pericycloid is also 
an epicycloid ; but there are three distinct kinds of trochoidal curves." As 
it will later be shown that every peri-trochoid can also be generated as an 
epi-trochoid we can scarcely escape the conclusion that the name peritrochoid 
would also have been rejeoted by Prof. Cliiford, had he been familiar with this 
property of double generation as belonging to the curtate and prolate fonns 
as well. But it is this very property, possessed also by the Jiyjm-tTOchoids, 
which necessitates a more extended nomenclature than that heretofoi-e 
existing, and I am not aware that there has been any attempt to 
provide the nine terms essential to its completeness. These it is my principal 
object to present, and that they have not before been suggested I attribute 
to the fact that the double generation of curtate and prolate trochoidal 
curves does not seem to have been generally known, being entirely ignored 
in many treatises which make c[uite prominent the fact that it is a property 
of the epi- and hj-po-cycloids, while, as far as I have seen, the few writers 
who mention it prove it indirectly, by showing the identity of trochoids with 
epicyclics and establishing it for the latter. 

As it is upon this peculiar and interesting feature that the nomen- 
clature, as now completed, depends, it may be well to give the demonstra- 
tions necessary to establish it. 

For the epi- and hj-po-cycloid probably the simjilest method of proof is 
that of Duhamel and which we may call a kinematic, as distinguished from 
a strictly geometrical, demonstration. It is, in substance, as follows : — 





Let F (Figs. 1 and 2) be the centre of the fixed cii-cle, and r that of a 
i-ollmg circle, the tracing point, P, being in the circumference of the latter. 



8 



The point of contact, 3, is — at the moment that the circles are in the rela- 
tive position indicated — an instantaneous centre of rotation for every point 
in the plane of the rolling circle ; the line, P^, joining such point of contact 
with the tracing point is therefore a normal to the trochoid that the point P 
is tracing. But if the normal Pg be produced to intersect the fixed circle 
in a second point, Q, it is evident that the same infinitesimal arc of the 
trochoid would be described with Q serving as instantaneous centre as when 
q fulfilled that office. The point, P, will, therefore, evidently trace the same 
curve, whether it be considered as in the circumference of the circle r, or in 
that of a second and larger circle, R, tangent to the fixed circle at Q. 

It is worth while, in this connection, to note what erroneous ideas with 
regard to these same loci were held by some writers as late as the middle of this 
century, — ideas whose falsity it would seem as if the most elementary geometri- 
cal construction would have exposed. Reuleaux instances the following state- 
ment made by Weissenborn in his Cyclischen Kurven (1856) : " If the circle 

g' described about rtio roll upon that 

^g" described about M, and if the describ- 
ing point, Bo, describe the curve 
BoPjPj as the inner circle rolls upon 
the arc Bob, then, evidently, if the 
smaller circle be fixed and the larger 
one rolled upon it in a direction o])- 
posite to that of the former rotation, 
the jjoint of the great circle which at 
the beginning of the operation coin- 
cided with Bo describes the same line 
BoPiP,." The fallacy of this state- 
ment is to us, perhaps, in the light 
of what has preceded, a Httle more 
evident than Weissenborn's deduc- 
tion ; although, as Reuleaux says, 
"his ' evidently ' expresses the usual notion, and the one which is suggested 
by a hasty pre-judgment of the case. In point of fact Bo describes the peri- 
cycloid BoB'B'', which certainly differs sufficiently from the hypocycloid 

BoP^P," 

We have next to consider the curtate and prolate epi-, hypo- and peri- 

trochoids. 

As previously stated, I have seen no direct proof that they also possess 
the same property of double generation, but find the kinematic method 
equally applicable to them and preferable on account of its simplicity. [It 
may safely be assumed that all present are so familiar with the ordinary 





DOUBLE GENERATION OF HTPOTROCHOIDS. 



10 

method of constructing trochoids as to render any explanation of it unnec- 
essary. But as I am expecting to make future use of this paper with some 
who are but beginners* in the study of these loci, I have given iu each figure 
the geometrical construction of that portion of the curve traced during the 
turning of each generator through an angle of 180° about its own centre.] 

For the hypotrochoids, let R, Fig. 4, be the centre of the first rolling- 
circle or generator, F that of the first director and P the initial position of 
the tracing point. The initial point of tangency of generator and director 
is TO. Let the generator roll over any arc of the director, as toQ. The cen- 
tre, E will then be found at K^ and the tracing point P at P^. The point of 
contact, Q, will then be the instantaneous centre of rotation for P2, and P2Q 
will, therefore, be a normal to the trochoid for that particular position of 
the tracing point. 

The motion of P is evidently circular about E, while that of E is in a 
circle about F. The curve PPiPj- ■ . . . Pg is that portion of the hypo- 
trochoid which is described while P describes an arc of 180° about E, the 
latter meanwhile moving through an arc of 108° about F, the ratio of the 
radii being 3:5. 

Now while tracing the curve indicated, the point P can be considered 
as rigidly connected with a second point, p, about which it also describes a 
circle, p meanwhile (like E) describing a circle about F. Such a point may 
be found as follows. Take any position of P, as P.^, and join it with the cor- 
responding position of E, as Ej ; also join E2 to F. Let us then suppose 
PjE and EjF to be adjacent links of a four-link mechanism. Let the 
remaining links, Fp^ and p^P.^, be parallel and equal to PjEj and E^F respec- 
tively. It is then evident that in the turning of E.^ about F, and of 
Pj about E.^, the point p^ will describe a circle about F, and that the 
motion of V.^ with respect to p,^ will be in a circular arc of radius p-^Pj. We 

* Such will need the following explanation of Fig. 4 : 

It must be evident that, in the rolling of the flist generator ujDon its director, the cen- 
tre R of the former will desorihe a circle ahout F and that the tracing point P will first 
approach F and then recede from it; also that P will Ije nearest to F when it has turned 
18U°al)out the centre R. This will occur when 180° of the circumference of the generator 
has come in contact with that of the director. But 108° of arc of the latter is equal in 
length to 180° of the former, equal arcs on unequal circles being subtended by angles at 
the centre which are inversely proportional to the radii. Drawing throughFaline mak- 
ing an angle of 108° with the line Fm which joins F with the initial point of contact of the 
rolling and fixed circles, we find P upon It at Pc, It meanwhile having reached Rj on the 
same line. 

To find intermediate positions of P divide Into the same number of equal parts both 
the path of centres R-Rj and the semi-circumference P-VI drawn with radius RP. In the 
figure six equal divisions have been taken, giving us Ri Ro. . . . Ro on the former and I, 
II. . . . VI on the latter. With Fas a centie draw arcs through the points I, II, III, &c., in 
succession. 

After turning one-sixth of 180° about R, i. e. through the angle PRI, the point P will 
evidently be nearer to F by the difference between FP and FI and somewhere on the arc 



11 

may therefore with equal correctness consider p., as the centre of a generator 
carrying tlie point P.„ and p,F a new line of centres, intersected by the normal 
PjQ in a second instantaneous centre, q, which, in strictest analogy with Q, 
divides the line of centres on which it lies into segments, p.,q aud Fq, which 
are the radii of the second generator and director respectiyely ; q being (like 
Q) the point of contact of tlie rolling and fixed circles for the instant that the 
tracing point is at P,^. The second generator and director, having p.^q and q¥ 
respectively for their radii, are represented in their initial iiositions, p being 
the centre of the fomier and /x the initial point of contact. The second 
generator rolls in the opposite direction to the first. 

It is important to notice that whereas the tracing point is in the first 
case within the generator and therefore traces the curve as a prolate hypo- 
trochoid, it is ivithout the second generator and describes the same curve as 
a curtate hypotrochoid. If we now let E, and F denote no longer the cen- 
tres, but the radii, of the rolling and fixed circles, respectively, we have for 
the fu'st generator and director 2R>F and for the second 2R<F. 

It has occurred to me that a distinction can very easily be made between 
trochoids generated under these two opposite relations of radii, by using the 
simple and suggestive term major hypotrochoid when 2E, is greater than F, 
and minor hypotrochoid when the opposite relation prevails. We would 
xhen say that the preceding demonstration had established the identity of a 
major prolate with a minor curtate hypotrochoid. 

Similarly the identity of major curtate and minor prolate forms could 
be shown. 

If the tracing point were on the circumference of the generator the 
trochoids traced would be, by the new nomenclature, major and minor hypo- 
cycloids. 

It is worth noticing that for both hypo-cycloids and hypo-trochoids the 
centre F is the same for both generations, and that the radius F is also con- 
stant for both generations of a hypo-cycloid but variable for those of a hypo- 

drawn through I, with centre F. It will ulsobe at a distance IIP from K, which is now at 
Ri, having turned through one-sixth of R— Rj. 1" will therefore he at Pi, the intersection 
of arc throiigh I and an arc of radius PR and centre Rj. Successive positions of P can be 
similarly found. 

If P be considered as carried by the second generator the construction is In every par- 
ticular analogous to that just given. We And 72" as the valueof the angle mFo to be set oil' 
at the centre of the second director in order to cut olf an arc on its circumference equal to 
180° of the new generator, and especial attention is called to the fact that the angle mFo 
will always be the supplement of the angle mFn. The path of centres p—po is as before, 
divided into six equal parts, that number of divisions being taken merely for conveni- 
ence, since we can also divide into six equal parts the sonii-circumfcrence drawn with 
radius pP by simply continuing the arcs having centre F and passing through I, II, III, 
&c., until they intersect it at 1, 2, 3, &c. This last is the most interesting feature of the 
geometrical construction, yet one which, evidently, could not be otherwise if the rolling 
of two different generators is to give the same curve. 



12 

trochoid. Having given tlie radii of generator and director for the construc- 
tion of a hypo-trochoid, the method just illustrated will always give the 
lengths of the radii of the second rolling and fixed circles. The accuracy of 
the values thus obtained may be checked by simple formulae derived as 
follows : — 

Radii being given for generation as a major hypotrochoid, to find cor- 
responding values for the identical minor hypotrochoid. 

Let Fj denote the radius FQ [= Fm] of the first director. 
" Fj " " " F^ [= F/x] " " second " 
" r " " " E^Q [= Urn] " " first generator. 
" p " " " p.^q [= pfi] " " second " 
" tr " " tracing radius of the first generation, i. e., the dis- 
tance lijPj (or RP) of tracing point from centre of first generator. 

Let tp then equal the second tracing radius = p^P^ = pP. From 
similar triangles QFq and QR^Pj we have — 

Fj : Fj : : ^r : r 

whence F, = ^U^} (1) 

r 

alsop = F, - tr = ^lA^ _ tr = tr I -' - 1 | ... (2) 

and tp = pjP, =FR2 = d, the distance between the centers of first 
generator and director (3) 

If the radii be given for a minor hypotrochoid, we have — 

FQ : p.'P^ :: Fq : p,q 

from which we have, as before, 

„ , ,. 7.7 radius of qiven fixed circle X qiven tracinq radius ,.^ 

fixed radius desired = =^-^ -f ^ ^ . . . (4) 

radius of given generator 

and, similarly, formulae (2) and (3) give the radius of desired generator and 
the corresponding tracing radius. 

With the tracing point on the circumference of the generator, if we let 
R = radius of the latter for a major hypocycloid and r correspondingly 
for the minor curve, then 

for major hypocycloid R = F — r (5) 

"minor " r = F — R (6) 

For the curves intermediate between the major and minor hypotrochoids, 
viz., those traced when the diameter of the rolling circle is exactly half that 
of the fixed circle, a separate division seems essential to completeness, and 
for such I suggest the general name medial hypotrochoids. For these the 
formulae for double generation are the same as for the " major " and "minor " 
curves and similarly derived. 




JUl 



DOUBLE GENERATION OF EPI- AND PERI-TROCHOIDS. 



14 

With the tracing point on the circumference of the generator these 
curves reduce to straight lines, diameters of the director. In all other cases 
they are an interesting exception to what we might naturally expect, being 
neither looped nor wavy, but ellipses. The failure of the terms " looped " and 
"wavy " to apply to these medial curves is paralleled by tha.t of the adjectives 
"curtate" and " prolate," since — contrary to the signification of the latter 
terms — any ellipse generated as a curtate curve is larger than the largest 
prolate elliptical hypotrochoid having the same director. And as we have seen 
that, with scarcely an exception, " curtate" and "j'l'ol^te" apply equally to the 
same curve, our only reason for retaining them is the fact of their general 
acceptation as indicative of the location of the tracing point with respect 
to the circumference of the rolling circle. 

Since the medial hyj^otrochoids are either straight lines or ellipses, we can 
readily find for them that which we have found it useless to attempt to 
construct for the other trochoidal curves, viz., simple terms suggestive of 
their /orm; in fact the names " straight hypocycloid" and " elliptical hypo- 
trochoid" have long been familiar to us all and we have but to incorjjorate 
them into the nomenclature we are constructing. 

It only remains to show (Fig. 5) that a prolate epi-trochoid can be gen- 
erated as a curtate ^eW-trochoid, and vice versa, for which the demonstra- 
tion is analogous to that given for the hypo-curves and leads to the follow- 
ing formulae, derived from the similar triangles QFg and QRiP, (the values 
being supposed to be given for the epi-trochoid and desired for the peri- 
trochoid) : 

V = ?1M (7) 



2 

r 



P 



{? 



tr}h + l\ (8) 



tp =: d = distance between centres of given generator and 

director = F, + r (9) 

If given as a ^en- trochoid and desired as an epi-trochoid the tracing radius 
will again equal the distance between the given centres (in this case, how- 
ever := E — F) ; the formula for the radius of desired director will be of 
the same form as equations (1) and (7), but 

radius of second generator = tr ] 1 — -^ i (10) 

With the tracing point on the circumference of the generator, and letting R 
= radius of the same for a peritrochoid and r for an epitrochoid we have 

for the epicycloid, r = E ^- F (11) 

" the pericycloid, E = F + r (12) 



15 

Altliongh I have by no means discussed all the terms under which 
trochoidal curves have been previously known, yet enough has been said to 
indicate how annoying the inconsistencies and ambiguities of foimer names 
and definitions have been, and how desirable it is that some adequate and 
consistent system should be generally adopted. If it has been my good 
fortune to contribute materially towards reaching such a satisfactory con- 
clusion of the whole matter, if not, indeed, to have settled the question 
entirely, my object will have been accomplished. 

In conclusion I give the complete nomenclature in tabular form, the 
arrangement having been suggested by that of Kennedy and being both a 
modification and extension of his ingenious scheme. It does away, to a con- 
siderable extent, with the necessity for definition. 

The foregoing demonstrations have established the identity of the curves 
whose names are preceded by the same letter. 



TROCHOIDS. 



Position 

of 

Tracing 

or 

Describing 

Point. 


Circle rolling 

upon 
straight line. 


Circle rolling upon circle. 


External 
c )ntact. 


Internal contact. 


Linear 
Trochoids. 


Epilrochoids. 


Larger circle 
rolling. 


Smaller circle rolling. 


2 R > F* 


2 R< F* 


2 R=F* 


Pentrochoids. 


Major Hypotrochoids. 


Minor Hypotrochoids. 


Medial Hypotrochoids. 


On circumfer- 
ence of rolling 
circle. 


Cycloid. 


(a)Epicycloid. 


(a) Pericycloid. 


(d) Major Hypocycloid. 


(d) Minor Hypocycloid. 


Straight 
Hypocycloid. 


Within 
Circumference. 


Prolate 
Trochoid. 


(b) Prolate 
Epitrochoid. 

(c) Curtate 
Epitrochoid. 


(c) Prolate 
Peritrochoid. 


(e) Major Prolate 
Hypotrochoid. 


(f) Minor Prolate 
Hypotrochoid. 


(g) Prolate Elliptical 
Plypotrochnid. 


Without 
Circumference. 


Curtate 
Tr.ichoid. 


(b) Curtate 
Peritrochoid. 


(f) Major Curtate 
Hypotrochoid. 


(e) Minor Curtate 
Hypotrochoid. 


(g) Curtate Elliptical 
Hypotrochoid. 



*i?=ra(ii ws of rolling circle, F of fixed circle. 




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